1.1.1 Mediation
Design and evaluation of multi-component interventions
Better understanding of causal ordering of the variables in time
Mediation is a model of process and processes unfold over time, so mediation is inherently a longitudinal model
Several different ways that X, M, and Y can exist in time
Cross-sectional: \(X_1 \rightarrow M_1 \rightarrow Y_1\)
Semi-longitudinal:
\(X_1 \rightarrow M_1 \rightarrow Y_2\)
\(X_1 \rightarrow M_2 \rightarrow Y_2\)
\(X_1 \rightarrow M_1 \rightarrow Y_3\)
\(X_1 \rightarrow M_3 \rightarrow Y_3\)
Maxwell & Cole (2007) and Maxwell, Cole, & Mitchell (2011)
Cross-sectional mediation almost always produces biased estimates of longitudinal mediation effects
Cross-sectional mediation effects may be higher OR lower than the longitudinal effects
The reason for this should be fairly obvious
Take-home message: If you want to know about longitudinal effects, don’t use cross-sectional data - use longitudinal data!
The mediated effect is the effect of X on Y via M
In SEM, such a path is described as the product of the regression coefficients that go into it
The a coefficient reflects the X \(\rightarrow\) M path
The b coefficient reflects the M \(\rightarrow\) Y path
The mediated effect is a \(\times\) b
MacKinnon et al. (2002), MacKinnon et al. (2004)
Joint significance: best balance of type I error and statistical power across conditions (sample size, effect size)
Product of coefficients: pretty good but difficult to actually use until PRODCLIN
Bootstrap: better confidence intervals than most other methods, requires programming skill or use of additional program, very flexible for more complex designs
Monte Carlo: better confidence intervals than most other methods, requires programming skill or use of additional program, very flexible for more complex designs
Proportion mediated = \(\frac{ab}{c} = 1 - \frac{c'}{c} = \frac{ab}{c' + ab}\)
Ratio of mediated to direct effect = \(\frac{ab}{c'}\)
Standardized mediated effect = \(\frac{ab}{SD_Y}\)
See Miočević et al. (2018) for comparisons
Proportion and ratio measures require very large samples and show bias in many situations
Proportion can be negative if direct and indirect effects are opposite sign
Standardized effect behaves well across sample sizes, effects
Also some \(R^2\) measures: see Lachowicz et al. (2018)
Preacher, K. J., & Kelley, K. (2011). Effect size measures for mediation models: quantitative strategies for communicating indirect effects. Psychological methods, 16(2), 93.
Miočević, M., O’Rourke, H. P., MacKinnon, D. P., & Brown, H. C. (2018). Statistical properties of four effect-size measures for mediation models. Behavior research methods, 50(1), 285-301.
Lachowicz, M. J., Preacher, K. J., & Kelley, K. (2018). A novel measure of effect size for mediation analysis. Psychological Methods, 23(2), 244.
\(a \times b = 0.584 \times 0.310 = 0.181\)
\(c - c' = 0.311 - 0.130 = 0.181\)
Joint significance: \(a\) is significant, \(b\) is significant
PRODCLIN via web: 95% CI = [-0.013, 0.489]
Bootstrap (lavaan): 95% CI = [-0.019, 0.488]
Monte Carlo program: 95% CI = [-0.019, 0.492]
Proportion mediated = \(\frac{ab}{c} = \frac{0.181}{0.311} = 0.582\)
Ratio of mediated to direct effect = \(\frac{ab}{c'} = \frac{0.181}{0.130} = 1.392\)
Standardized mediated effect = \(\frac{ab}{SD_Y} = \frac{0.181}{4.729} = 0.038\)
Temporal precedence
Timing of change is accurately measured
M and Y are normally distributed*
No additional influences have been omitted: confounders
The relationships are causal
X comes before M
M comes before Y
Mediation is a causal chain, so we expect that the links occur in order
Timing of measurement is important
Frequency
More rapid / fluctuating change requires more measurements
Nyquist criteria in EEG: sample at double the frequency to avoid aliasing
Timing
Have you allowed time for change to occur?
But measure before change fades
Timmons, A. C., & Preacher, K. J. (2015). The importance of temporal design: How do measurement intervals affect the accuracy and efficiency of parameter estimates in longitudinal research?. Multivariate behavioral research, 50(1), 41-55.
This is for standard methods of mediation analysis
For binary / count outcomes, see Geldhof et al. (2018)
Causal mediation models are also more flexible with this assumption
Last two assumptions combine into “two part sequential ignorability”
Assume that there are no additional influences affecting the relationships (i.e., no confounders of X, M, or Y), which is easy to do for randomized predictors
X can be randomized: no confounders if it is
M is never randomized, essentially turning the right half of the model into an observational study (even if X were randomized)
Can you make causal statements about the mediated effect?
Randomization is the gold standard for establishing causality
When people are randomized to condition, there should be no differences between the groups on any measured or unmeasured variables
Any differences between groups must be due to the manipulation
But there are many situations where randomization is not feasible or ethical
Two approaches: methodological and statistical
Logic of determining causation: Hill (1965)
Strength: stronger vs weaker relationship
Consistency: consistency by multiple people in multiple samples
Specificity: specific findings (i.e, specific disease vs general health)
Temporality: “cause” occurs prior to “effect”
Biological gradient: larger effect with larger exposure to “cause”
Plausibility: plausible and sensible mechanism
Coherence (agreement): agreement between laboratory and observational studies
Experiment: experimental evidence
Analogy: similar “causes” result in similar “effects”
Potential outcomes framework
Counter-factual:
You were assigned to condition X = 0
What would your Y value be if you were in condition X = 1 instead?
Also accounts for
Confounders of X, M, and Y
XM interaction on Y: Does the b path vary depending on X?
MacKinnon, D. P., Valente, M. J., & Gonzalez, O. (2020). The correspondence between causal and traditional mediation analysis: The link is the mediator by treatment interaction. Prevention Science, 21(2), 147-157.
Rijnhart, J. J., Lamp, S. J., Valente, M. J., MacKinnon, D. P., Twisk, J. W., & Heymans, M. W. (2021). Mediation analysis methods used in observational research: a scoping review and recommendations. BMC medical research methodology, 21(1), 1-17.
Valente, M. J., Rijnhart, J. J., Smyth, H. L., Muniz, F. B., & MacKinnon, D. P. (2020). Causal mediation programs in R, M plus, SAS, SPSS, and Stata. Structural equation modeling: a multidisciplinary journal, 27(6), 975-984.
Older, more technical sources
Judea Pearl, Tyler Vanderwheele
The last slide showed a prospective model
It seems like it’s longitudinal because there are 3 times points
But it’s NOT because there’s no change in any variable
X is only at time 1, M is only at time 2, Y is only at time 3
The prospective model does not convey actual change in a variable over time
We can make some modifications to the model to include change
a path: Relationship between X1 and M2
(No change in either X or M)
b path: Relationship between M2 and Y3
(No change in either M or Y )
There are two simple ways to turn a prospective model into a longitudinal model
There are also some more complex ways and methods to incorporate mediation into growth models
a path: \[\hat{M}_{2-1} = i_{MX} + aX_1\]
b and c’ paths: \[\hat{Y}_{3-2} = i_{YXM} + bM_{2-1} + c'X_1\]
c path: \[\hat{Y}_{3-2} = i_{YX} + cX_1\]
a path: Relationship between X1 and the absolute change in M from time 1 to time 2
b path: Relationship between the absolute change in M from time 1 to time 2 and absolute change in Y from time 2 to time 3
The mediated effect reflects the effect of X on the change in Y from time 2 to 3, via the change in M from time 1 to time 2
Similar strengths & weaknesses to the 2-wave difference score models
Difference scores work well if there are few pre-test differences
a path: \[\hat{M}_2 = i_{MX} + aX_1 + dM1\]
b and c’ paths: \[\hat{Y}_3 = i_{YXM} + bM_2 + c'X_1 + eY_2\]
c path: \[\hat{Y}_3 = i_{YX} + cX_1 + fY_2\]
a path: Relationship between X1 and the average change in M from time 1 to time 2
b path: Relationship between the average change in M from time 1 to time 2 and average change in Y from time 2 to time 3
The mediated effect reflects the effect of X on Y3 (controlling for Y2) via M2 (controlling for M1)
Similar strengths & weaknesses to the 2-wave ANCOVA
As with 2 wave models, ANCOVA works well if you have pre-test differences that make sense to equate across groups
Concern:
These models are very complicated, require SEM software to run
Time is treated as discrete and equally spaced
Auto-regressive models tend to focus more on the “stability” aspect of the model, while many of our research questions focus on the “change” aspect
The cross-lag relations among variables are often inaccurate
Cheong, J., MacKinnon, D. P., & Khoo, S. T. (2003). Investigation of mediational processes using parallel process latent growth curve modeling. Structural Equation Modeling, 10(2), 238-262.
Basic extension of what we’ve alredy talked about with parallel process models with regression paths
But now we can talk about the indirect effects too
X = parent substance use
M = growth model of cigarette use (centered at age 16)
Y = growth model of alcohol use (centered at age 19)
Thinking about mediation:
X is temporally before M – measured at same time but X is about past behavior
M is temporally before Y – at least, the intercepts are
## lavaan 0.6-10 ended normally after 83 iterations
##
## Estimator ML
## Optimization method NLMINB
## Number of model parameters 27
## Number of equality constraints 8
##
## Number of observations 749
## Number of missing patterns 6
##
## Model Test User Model:
##
## Test statistic 96.087
## Degrees of freedom 56
## P-value (Chi-square) 0.001
##
## Parameter Estimates:
##
## Standard errors Standard
## Information Observed
## Observed information based on Hessian
##
## Latent Variables:
## Estimate Std.Err z-value P(>|z|)
## i_alc =~
## alcuse15 1.000
## alcuse16 1.000
## alcuse17 1.000
## alcuse18 1.000
## alcuse19 1.000
## s_alc =~
## alcuse15 -4.000
## alcuse16 -3.000
## alcuse17 -2.000
## alcuse18 -1.000
## alcuse19 0.000
## i_cig =~
## ciguse15 1.000
## ciguse16 1.000
## ciguse17 1.000
## ciguse18 1.000
## ciguse19 1.000
## s_cig =~
## ciguse15 -1.000
## ciguse16 0.000
## ciguse17 1.000
## ciguse18 2.000
## ciguse19 3.000
##
## Regressions:
## Estimate Std.Err z-value P(>|z|)
## i_cig ~
## paruse (a1) 0.080 0.014 5.711 0.000
## s_cig ~
## paruse (a2) 0.009 0.007 1.227 0.220
## i_alc ~
## i_cig (b1) 0.187 0.068 2.753 0.006
## s_cig (b2) 1.543 0.452 3.412 0.001
## s_alc ~
## i_cig (b3) -0.028 0.020 -1.412 0.158
## s_cig (b4) 0.508 0.144 3.515 0.000
## i_alc ~
## paruse (cp1) 0.020 0.023 0.863 0.388
## s_alc ~
## paruse (cp2) -0.001 0.007 -0.092 0.927
##
## Covariances:
## Estimate Std.Err z-value P(>|z|)
## .i_alc ~~
## .s_alc 0.047 0.039 1.206 0.228
##
## Intercepts:
## Estimate Std.Err z-value P(>|z|)
## .alcuse15 0.000
## .alcuse16 0.000
## .alcuse17 0.000
## .alcuse18 0.000
## .alcuse19 0.000
## .ciguse15 0.000
## .ciguse16 0.000
## .ciguse17 0.000
## .ciguse18 0.000
## .ciguse19 0.000
## .i_alc 4.828 0.466 10.368 0.000
## .s_alc 0.296 0.138 2.148 0.032
## .i_cig 5.507 0.173 31.760 0.000
## .s_cig 0.017 0.093 0.178 0.858
##
## Variances:
## Estimate Std.Err z-value P(>|z|)
## .alcuse15 (r1) 0.392 0.023 17.275 0.000
## .alcuse16 (r1) 0.392 0.023 17.275 0.000
## .alcuse17 (r1) 0.392 0.023 17.275 0.000
## .alcuse18 (r1) 0.392 0.023 17.275 0.000
## .alcuse19 (r1) 0.392 0.023 17.275 0.000
## .ciguse15 (r2) 0.402 0.025 16.267 0.000
## .ciguse16 (r2) 0.402 0.025 16.267 0.000
## .ciguse17 (r2) 0.402 0.025 16.267 0.000
## .ciguse18 (r2) 0.402 0.025 16.267 0.000
## .ciguse19 (r2) 0.402 0.025 16.267 0.000
## .i_alc 0.377 0.130 2.899 0.004
## .s_alc 0.024 0.014 1.756 0.079
## .i_cig 1.176 0.078 15.005 0.000
## .s_cig 0.079 0.019 4.050 0.000
##
## Defined Parameters:
## Estimate Std.Err z-value P(>|z|)
## a1b1 0.015 0.006 2.481 0.013
## a1b3 -0.002 0.002 -1.371 0.170
## a2b2 0.014 0.012 1.153 0.249
## a2b4 0.005 0.004 1.151 0.250